Redundancy — cold standby, hot standby, active redundancy: Foundations

3.6.32 · D1 · Physics › Spacecraft Structures & Systems Engineering › Redundancy — cold standby, hot standby, active redundancy

Parent note on cold/hot/active redundancy padhne se pehle, tumhe har woh symbol samajhna hoga jo usne zero se introduce kiya hai. Hum ek ek brick se chalenge. Neeche kuch bhi yeh assume nahi karta ki tumne pehle probability, exponentials, ya calculus dekha hai.

1. Reliability $R (t)$ — survive karne ka chance

Picture: $1000$ identical radios ko ek saath on karo. Ghante guzarne ke saath, kuch band ho jaate hain. Agar $5$ ghante baad $900$ abhi bhi kaam kar rahe hain, toh $R (5 hours) = 900/1000 = 0.9$ .

Figure — Redundancy — cold standby, hot standby, active redundancy

Topic ko yeh kyun chahiye: har dawa jaise "do units ek se zyada reliable hain" basically $R (t)$ ke baare mein ek dawa hai. Figure mein curve dekho — yeh $1$ se start hota hai (sabhi $t = 0$ par zinda) aur $0$ ki taraf slide karta hai. Redundancy ek kalaa hai is curve ko daayein dhakelne ki taaki woh zyada der tak upar rahe.

2. Failure rate $λ$ — cheezein kitni tezi se tootengi

Picture: $λ$ ko death slide ki steepness ki tarah socho. Ek sasta part steep slide rakhta hai (bada $λ$ , jaldi marta hai); ek space-grade part gentle slide rakhta hai (bahut chhota $λ$ ).

Yeh tool kyun aur doosra kyun nahi? Hum ek constant $λ$ use karte hain kyunki, electronics ke useful mid-life mein, survivors ke beech risk roughly steady rehta hai — ek chip jo 1000 ghante chali hai woh apne agle ghante mein ek nayi chip se zyada marne ki sambhavna nahi rakhti. Woh "memoryless" behaviour exactly wahi hai jo constant $λ$ capture karta hai, aur yahi cheez parent note ke saaf formulas ko possible banati hai.

3. Exponential $e^{- λ t}$ — survival is shape kyun decay karta hai

Picture: yeh Section 1 ki death slide hai, formula mein likhi hui. Neeche figure dekho jahan bada $λ$ (coral) tezi se girta hai aur chhota $λ$ (mint) dheere dheere girta hai.

Topic ko yeh kyun chahiye: parent mein har reliability formula — $R_{hot}$ , $R_{cold}$ , $R_{TMR}$ — $e^{- λ t}$ ki copies combine karke bana hai. Iss ek building block ko master karo aur baaki sab arithmetic hai.

Recall Quick check: kisi bhi component ke liye

$R (0)$ kya hai? $R (0) = e^{- λ \cdot 0} = e^{0} = 1$ . Time zero par sab zinda hain. ::: $R (0) = 1$

4. "Fail ho gaya hai ka probability" — complement $1 - e^{- λ t}$

Topic ko yeh kyun chahiye: hot standby tabhi fail hota hai jab dono units dead hon. "Dono dead" compute karne ke liye pehle "ek dead hai" chahiye, jo exactly yeh complement hai. Yeh phrase pakad ke rakho: dead-chance $= 1 - e^{- λ t}$ .

5. Independent chances combine karna — multiply karo

Picture: socho $10 \times 10$ outcomes ka grid. Agar A $30%$ columns mein fail hota hai aur B $30%$ rows mein fail hota hai, toh "dono fail" corner $0.3 \times 0.3 = 0.09$ poore square ka hai.

Yeh tool kyun aur addition kyun nahi? Hum multiply karte hain, add nahi, kyunki hum overlap chahte hain ("A dead AND B dead"), total nahi ("A dead OR B dead"). Yeh hot-standby derivation mein sabse important step hai: system reliability $= 1 - (both dead) = 1 - (1 - e^{- λ t})^{2}$ .

6. MTTF aur integral $\int_{0}^{\infty} R (t) d t$

Picture: figure mein shaded lavender region hi MTTF hai. Jo curve zyada der tak upar rehta hai woh zyada area enclose karta hai — zyada average life. Yahi redundancy hai, geometrically.

Death slope jitna gentle (chhota $λ$ ), average life $1/ λ$ utna lamba — jo intuition se perfectly match karta hai.

7. Outputs mein se choose karna — voting aur majority

Picture: teen arrows ek value ki taraf point karte hain; do "5" ki taraf point karte hain, ek "9" ki taraf. Voter "5" pick karta hai aur outlier throw away karta hai. Ek jhoota do sachon ko outvote nahi kar sakta.

Topic ko yeh kyun chahiye: active redundancy (Triple Modular Redundancy) kuch bhi kabhi switch off nahi karta — woh simply galat output ko out-vote karke mask karta hai. Yeh standby se alag survival strategy hai, aur yeh Byzantine Fault Tolerance se connect hota hai jab ek faulty unit inconsistently jhooth bolta hai, aur Single Event Upset (SEU) se as ek common cause of single wrong bit.

Prerequisite map

Upar har arrow ek "pehle yeh samjho phir woh" hai. Koi bhi path trace karo aur tum parent note upar se neeche padh sakte ho.

Related building blocks in hain Power Budget (isliye hot standby ka always-on cost kyun matter karta hai), Failure Modes and Effects Analysis (FMEA) aur Fault Detection and Isolation (failure kaise notice hoti hai switchover hone se pehle). Parent topic hai the redundancy note.

Equipment checklist

Daayein side cover karo aur parent note kholne se pehle har ek ka jawab do.

$R (t)$ ka plain words mein kya matlab hai? ::: Woh probability ki ek component time $t$ tak run karne ke baad abhi bhi kaam kar raha hai; $0$ se $1$ ke beech ki ek number.
Time variable $t$ kya range leta hai? ::: $t \geq 0$ — launch se elapsed running time, kabhi negative nahi.
$λ$ precisely kya hai? ::: Instantaneous hazard rate — currently surviving units ka woh fraction jo per hour marta hai, total failures per hour ki plain count nahi.
$d t$ ka kya matlab hai? ::: Time ka ek infinitesimally small slice (ek duration jo zero ki taraf shrink ho raha hai).
Single-unit reliability ko kaun sa differential equation govern karta hai, aur uska solution kya hai? ::: $d R / d t = - λ R$ , jiska solution $R (t) = e^{- λ t}$ hai.
Kisi bhi unit ke liye $R (0)$ kya hai? ::: $1$ — launch par abhi tak kuch bhi fail nahi hua.
Time $t$ tak ek unit ke fail ho jaane ka chance kya hai? ::: $1 - e^{- λ t}$ , survival chance ka complement.
Hum do units ke liye $(1 - e^{- λ t})$ ko khud se kyun multiply karte hain? ::: "Dono dead" paane ke liye — independent events ke liye multiply rule dono ke fail hone ka overlap deta hai.
$\int_{0}^{\infty}$ mein $\infty$ ka kya matlab hai? ::: Koi number nahi, balki "let $t$ hamesha ke liye badhe"; safe hai kyunki $e^{- λ t}$ itni tezi se decay hota hai ki total finite rehta hai.
$\int_{0}^{\infty} R (t) d t$ kya compute karta hai, aur woh kaisa dikhta hai? ::: MTTF (average lifetime); geometrically yeh survival curve ke neeche ka area hai.
Ek single unit ke liye MTTF kya hai, aur integral kaise ki jaati hai? ::: $1/ λ$ , $\int_{0}^{\infty} e^{- λ t} d t = [- \frac{1}{λ} e^{- λ t}]_{0}^{\infty} = 1/ λ$ se.
$(k n)$ ka kya matlab hai aur uska formula kya hai? ::: $n$ items mein se $k$ choose karne ke tareekon ki sankhya; $(k n) = \frac{n !}{k ! ( n - k )!}$ .
majority( $A, B, C$ ) kya karta hai? ::: Woh output karta hai jo teen units mein se kam se kam do agree karte hain, outlier ko ignore karta hai.